Question

Show that the adiabatic bulk modulus, defined as \(B=-V(d P / d V),\) for an ideal gas is equal to \(\gamma P\).

Short answer

In order to show that the adiabatic bulk modulus for an ideal gas is equal to γP, we used the ideal gas law and the relation for an adiabatic process for an ideal gas, which is PV^γ = constant. By differentiating both sides of this relation with respect to volume, applying the product rule and chain rule for differentiation, and rearranging the equation to find dP/dV, we obtained an expression for dP/dV in terms of P and V. Then, we substitute this expression into the definition of the adiabatic bulk modulus, which is B = -V(dP/dV), and simplified to obtain the result B = γP.

Step-by-step solution

Write down the relation for an adiabatic process

For an ideal gas undergoing an adiabatic process, \(PV^{\gamma} = \text{constant}\). Let's call this constant \(K\). So, $$PV^{\gamma} = K.$$

Differentiate both sides with respect to volume

We will differentiate both sides of the above equation with respect to \(V\): $$\dfrac{d(PV^{\gamma})}{dV} = \dfrac{dK}{dV}.$$ Since \(K\) is a constant, its derivative with respect to \(V\) is zero, so: $$\dfrac{d(PV^{\gamma})}{dV} = 0.$$ Note that to differentiate \(PV^{\gamma}\), we will use the product rule and the chain rule.

Apply the product rule and chain rule for differentiation

Applying the product rule and chain rule to the left side of the equation in Step 2, we get: $$\dfrac{dP}{dV}V^{\gamma} + P\dfrac{dV^{\gamma}}{dV} = 0.$$ Now we need to find the expression for \(dV^{\gamma}/dV\). Using the chain rule: $$\dfrac{dV^{\gamma}}{dV} = \gamma V^{\gamma - 1}.$$ Substitute this expression back into the equation from Step 2: $$\dfrac{dP}{dV}V^{\gamma} + P(\gamma V^{\gamma - 1}) = 0.$$

Rearrange the equation to find \(dP/dV\)

Let's isolate \(dP/dV\) in the equation above: $$\dfrac{dP}{dV}V^{\gamma} = -P(\gamma V^{\gamma - 1}).$$ Then divide both sides by \(V^{\gamma}\) to get: $$\dfrac{dP}{dV} = -\dfrac{P(\gamma V^{\gamma - 1})}{V^{\gamma}}.$$ Now cancel the terms with volume: $$\dfrac{dP}{dV} = -\gamma\dfrac{P}{V}.$$

Put the expression for \(dP/dV\) into the definition of \(B\)

Now we will substitute the expression we just derived for \(dP/dV\) into the definition of the adiabatic bulk modulus \(B = -V(dP/dV)\): $$B = -V\left(-\gamma\dfrac{P}{V}\right).$$

Simplify the expression for \(B\)

Now simplify the expression for \(B\): $$B = \gamma P.$$ Thus, we have shown that the adiabatic bulk modulus for an ideal gas is equal to \(\gamma P\).

Key concepts

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is transferred into or out of the system. The system is effectively insulated from its surroundings, causing changes in pressure and volume to occur without any heat exchange. For an ideal gas, this process is described by the equation \(PV^{\gamma} = K\), where \( P \) is the pressure, \( V \) is the volume, \( \gamma \) is the heat capacity ratio (which is the ratio of the heat capacity at constant pressure to the heat capacity at constant volume), and \( K \) is a constant specific to the particular adiabatic process. During an adiabatic compression or expansion, the temperature of the gas will change even though there is no heat flow. This characteristic is vital in understanding phenomena such as the operation of heat engines and atmospheric thermodynamics.

Ideal Gas

The concept of an ideal gas is a simplified model that allows us to analyze the behavior of gases under various conditions without getting bogged down in the complexities of real-world interactions. In this model, we assume that the molecules of an ideal gas do not attract or repel each other and that the volume of the gas molecules is negligible compared to the volume of the container. These assumptions lead to the ideal gas law, expressed as \(PV = nRT\), where \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in kelvin. The ideal gas law can be applied to many gas-related problems and serves as a fundamental building block for understanding more complex thermodynamic systems.

Product Rule

The product rule is a critical concept in calculus, especially when dealing with differentiation. It comes into play when you need to find the derivative of the product of two functions. The product rule states that the derivative of a product of two functions \( f(x) \) and \( g(x) \) is \( f'(x)g(x) + f(x)g'(x)\). Applying the product rule simplifies the process of differentiation when faced with the multiplication of variables, as is often encountered in physics and engineering problems. It provides the foundation for working out the derivatives needed to describe the behavior of various physical systems, including the gases involved in the adiabatic process.

Chain Rule

The chain rule is another essential tool in calculus, particularly in differentiation. It is used when taking the derivative of a composite function - essentially, a function of another function. The chain rule tells us that to find the derivative of a composite function \( h(x) = f(g(x)) \), we need to multiply the derivative of the outer function, \( f \) with respect to the inner function \( g \) by the derivative of the inner function \( g \) with respect to \( x \): \( h'(x) = f'(g(x))g'(x) \). This rule allows us to dissect complex expressions into simpler parts, making it easier to differentiate and analyze. In the context of an adiabatic process, the chain rule is pivotal when differentiating expressions involving powers of the volume, such as \( V^{\gamma} \).

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any point and is fundamental in calculus. When differentiating, we use rules like the product rule and the chain rule to break down complex expressions into more manageable terms for which derivatives can be more easily found. Differentiation has numerous applications in various fields, including understanding rates of change in science and engineering, finding maxima and minima in optimization problems, and analyzing motion in physics. In the case of adiabatic processes for gases, differentiation helps to determine the relationship between pressure and volume changes and thus allows for the calculation of properties like the adiabatic bulk modulus.